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* 23. 1.

Take a straight line DE given in position and magni

А tude, and at the points D, E,

D make * the angle EDF equal to the angle BAC, and the angle DEF equal to ABC; therefore the other angles

B

E EFD, BCA are equal, and each of the angles at the points A, B, C is given, wberefore each of those at the points D, E, F, is given: And because the straight line FD is drawn to the given point D in DE which is given in position, making the given angle EDF; therefore DF is given in position* ; * 32 Dat. In like manner EF also is given in position; wherefore the point F is given: and the points D, E are given; therefore each of the straight lines DE, EF, FD is given * in magnitude; wherefore the triangle DEF is • 29 Dat. given in species *; and it is similar * to the triangle ABC; which therefore is given in species.

1 Def. 6.

48 Dat.

* 4. 6.

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If one of the angles of a triangle be given, and if the

sides about it have a given ratio to one another ; the triangle is given in species.

Let the triangle ABC have one of its angles BAC given, and let the sides BA, AC, about it have a given ratio to one another; the triangle ABC is given in species.

Take a straight line DE given in position and magnitude, and at the point D, in the given straight line DE, make the angle EDF equal to the given angle BAC: wherefore the angle EDF is given ; and because the straight line FD is drawn to the given point D in ED, which is given in position, making the given angle EDF;

A therefore FD is given in posi

D tion *. And because the ratio

* 32 Dat. of BA to AC is given, make the ratio of ED to DF the same B С E E F with it, and join EF; and because the ratio of ED to DF is given, and ED is given, therefore * DF is given in magnitude: and it is given * 2 Dat. also in position, and the point D is given, wherefore the point F is given *; and the points D, E are given, * 30 Dat.

* 29 Dat.

42 Dat.

wherefore DE, EF, FD are given* in magnitude;'and the triangle DEF is therefore given* in species; and because the triangles ABC, DEF have one angle BAC equal to one angle EDF, and the sides about these angles proportionals; the triangles are * similar, but the triangle DEF is given in species, and therefore also the triangle ABC.

* 6. 6.

42.

PROP. XLV.

See N.

If the sides of a triangle have to one another given ratios ;

the triangle is given in species. Let the sides of the triangle ABC have given ratios to one another, the triangle A.BC is given in species.

Take a straight line D given in magnitude; and be

cause the ratio of AB to BC is given, make the ratio * 2 Dat.

of D to E the same with it; and D is given, therefore* E is given. And because the ratio of BC to CA is

given, to this make the ratio of E to F the same; and 2 Dat.

E is given, and therefore * F. And because as AB to
BC, so is D to E; by composition AB and BC to-
gether are to BC, as D
and E to F: but as BC

A
to CA, so is E to F;
* 22. 5. therefore, ex æquali *, as
AB and BC are to CA,

B4 so are D and Eto F, and

D E Ė AB and BC are greater

C than CA; therefore D * A. 5. and E are greater * than

K

Н. F. In the same manner, any two of the three D, E, F are greater than the third. Make * the triangle GHK, whose sides are equal to D, E, F, so that GH be equal to D, HK to E, and KG to F; and because D, E, F are, each of them, given, therefore GH, HK, KG are each of them

given in magnitude; therefore the triangle GHK is * 42 Dat. given * in species : but as AB to BC, so is (D to E,

that is) GH to HK; and as BC to CA, so is (E to F, that is) HK to KG; therefore, ex æquali, as AB to AC, so is GH to GK. Wherefore the triangle ABC is equiangular and similar to the triangle GHK; and the triangle GHK is given in species: therefore also the triangle ABC is given in species.

• 20.1.

* 22. 1.

* 5. 6.

Cor. If a triangle be required to be made, the sides of which shall have the same ratios which three given straight lines D, E, F have to one another; it is necessary that every two of them be greater than the third.

PROP. XLVI.

• 19. 1.

A. 5.

F

If the sides of a right angled triangle about one of the

acute angles have a given ratio to one another; the triangle is given in species.

Let the sides AB, BC about the acute angle ABC of the triangle ABC, which has a right angle at A, have a given ratio to one another; the triangle ABC is given in species.

Take a straight line DE given in position and magnitude; and because the ratio of AB to BC is given, make as AB to BC, so DE to EF; and because DE bas a given ratio to EF, and DE is given, therefore * .2 Dat. EF is given; and because as AB to BC, so is DE to EF; and AB is less * than BC, therefore DE is less * than EF. From the point D draw DG at right angles to DE, and from the centre E, at the distance EF, describe a circle which shall meet DG in

C two points; let G be B either of them, and join

E

G EG: therefore the circumference of the circle is given * in position; and the * 6 Def. straight line DG is given * in position, because it is * 32 Dat. drawn to the given point D in DE given in position, in a given angle; therefore * the point G is given, and * 28 Dato the points D, E are given, wherefore DE, EĞ, GD are given in magnitude, and the triangle DEG in .29 Dat. species * And because the triangles ABC, DEG, 42 Dat. have the angle BAC equal to the angle EDG, and the sides about the angles ABC, DEG proportionals, and each of the other angles BCA, EGD, less than a right angle; the triangle ABC is equiangular* and similar to * 7.6. the triangle DEG; but DEG is given in species; therefore the triangle ABC is given in species : and in the same manner, the triangle made by drawing a straight line from E to the other point in which the circle meets DG, is given in species.

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44.

* 32. 1.

PROP. XLVII. See N. If a triangle have one of its angles which is not a right

angle given, and if the sides about another angle have a given ratio to one another; the triangle is given in species.

Let the triangle ABC have one of its angles ABC a given, but not a right angle, and let the sides BA, AC, about another angle BAC have a given ratio to one another; the triangle ABC is given in species.

First, let the given ratio be the ratio
of equality, that is, let the sides BA, AC, А.
and consequently the angles ABC, ACB,
be equal; and because the angle ABC is
given, the angle ACB, and also the re-

maining * angle BAC is given; there* 43 Dat. fore the triangle ABC is given * in spe- B C

cies; and it is evident that in this case
the given angle ABC must be acute.

Next, let the given ratio be the ratio of a less to a
greater, that is, let the side AB adjacent to the given
angle be less than the side AC: take a straight line
DE given in position and magnitude, and make the
angle DEF equal to the given angle ABC: there-
fore EF is given * in posi-
tion; and because the ratio
of BA to AC is given, as
BA to AC, so make ED to
DG; and because the ratio

B4

C of ED to DG is given, and ED is given, the straight line DG is given *, and BA

is less than AC, therefore * A. 5. ED is less than DG. From

E
the centre D, at the distance
DG, describe the circle GF
meeting EF in F, and join

G
DF; and because the circle

is given * in position, as also 28 Dat. the straight line EF, the point F is given *; and the

points

D, E, are given; wherefore the straight lines DE, * 29 Dat. EF, FD are given * in magnitude, and the triangle

DEF in species *. And because BA is less than AC,

the angle" ACB is less * than the angle ABC, and * 17. 1. therefore ACB is less * than a right angle. In the

32 Dat.

2 Dat.

* 6 Dat.

* 42 Dat.

* 18. 1.

same manner because ED is less than DG or DF, the angle DFE is less than a right angle: and because the triangles ABC, DEF, have the angle ABC equal to the angle DEF, and the sides about the angles BAC, EDF, proportionals, and each of the other angles ACB, DEF, less than a right angle; the triangles ABC, DEF are * similar, and DEF is given in species, * 7.6. wherefore the triangle ABC is also given in species.

Thirdly, let the given ratio be the ratio of a greater tó a less, that is, let the side AB adjacent to the given angle be greater than AC; and, as in the last case, take a straight line DE given in position and magnitude, and make

A the angle DEF equal to the given angle ABC; therefore EF is given * in position: also draw

* 32 Dat.

B DG perpendicular to EF; therefore if the ratio of BA to AC be

D the same with the ratio of ED to the perpendicular DG, the triangles ABC, DEG, are similar*,

* 7.6. because the angles ABC, DEG, are equal, and DGE is a right E G angle: therefore the angle ACB is a right angle, and the triangle ABC is given in * 43 Dat. species.

But if, in this last case, the given ratio of BA to AC be not the saine with the ratio of ED to DG, that is, with the ratio of BA to the perpendicular AM drawn from A to BC; the ratio of BA to AC must be less * than the ratio of BA to AM, because AC is greater than AM. Make as BA to AC, so ED to DH; therefore the ratio of ED to DH is less than the ratio of (BA to AM, that is, than the ratio of) ED to DG : and consequently DH is greater * than

* 10.5. DG; and because BA is greater

* A. 5. than AC, ED is greater * than DH. From the centre D, at the

B. distance DH, describe the circle KHF which necessarily meets the straight line EF in two points, because DH is greater than DG, and less than DE. Let the circle meet EF in the points F, K, which are given, as was shewn in

* 8. 5.

D

H

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